Problem: Express $z_1=6[\cos(135^{\circ})+i\sin(135^{\circ})]$ in rectangular form. Express your answer in exact terms. $z_1=$
Answer: The Strategy A complex number in rectangular form, $z={a}+{b}i$, can be written in polar form as $z={r}[\cos{\theta}+i\sin{\theta}]$, where ${r}$ is the absolute value, or modulus, and ${\theta}$ is the angle, or argument. Therefore, ${r}$ and ${\theta}$ can be found using the following formulas: ${r}=\sqrt{{a}^2+{b}^2}$ $\tan{\theta}=\dfrac{{b}}{{a}}$ [How did we get these equations?] Similarly, a complex number in polar form, $z={r}[\cos{\theta}+i\sin{\theta}]$, can be written in rectangular form as $z={a}+{b}i$, using the following formulas: ${a}={r}\cos{\theta}$ ${b}={r}\sin{\theta}$ [How did we get these equations?] Finding $a$ For $z_1={6}[\cos{135^\circ}+i\sin{135^\circ}]$ : ${r}={6}$ ${\theta}={135^\circ}$ Therefore, we can find ${a}$ as follows. $\begin{aligned}{a}&={r}\cos{\theta} \\\\&={6}\cos{135^\circ} \\\\&={-3\sqrt{2}}\end{aligned}$ Finding $b$ $\begin{aligned}{b}&={r}\sin{\theta} \\\\&={6}\sin{135^\circ} \\\\&={3\sqrt{2}}\end{aligned}$ Summary $z_1={-3\sqrt{2}}+{3\sqrt{2}}i$